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The p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems

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The formal p-adic numbers

Yes there is: the formal locale of p-adic integer is simply defined as the projective limit of the $\mathbb{Z}/p^k\mathbb{Z}$ (as a pro-finite locale). So internally in any topos a continuous function …
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